Front. Public HealthFrontiers in Public HealthFront. Public Health2296-2565Frontiers Media S.A.10.3389/fpubh.2023.1101436Public HealthOriginal ResearchMathematical epidemiological modeling and analysis of monkeypox dynamism with non-pharmaceutical intervention using real data from United KingdomNgunguMercy1AddaiEmmanuel23AdenijiAdejimi4AdamUmar Muhammad5OshinubiKayode6*1Human Sciences Research Council (HSRC), Pretoria, South Africa2Department of Biomedical Engineering, College of Biomedical Engineering, Taiyuan University of Technology, Taiyuan, China3Department of Mathematics, Taiyuan University of Technology, Taiyuan, China4Department of Mathematics, Tshwane University of Technology, Pretoria, South Africa5Department of Mathematics, Federal University, Dutse, Nigeria6AGEIS Laboratory, University Grenoble Alpes, Saint Martin d'Hères, France
Edited by: Nicola Luigi Bragazzi, York University, Canada
Reviewed by: Necati Özdemir, Balıkesir University, Turkey; Olumuyiwa James Peter, University of Medical Sciences, Nigeria; Maryam Shafaati, Tehran University of Medical Sciences, Iran
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In this study, a mathematical model for studying the dynamics of monkeypox virus transmission with non-pharmaceutical intervention is created, examined, and simulated using real-time data. Positiveness, invariance, and boundedness of the solutions are thus examined as fundamental features of mathematical models. The equilibrium points and the prerequisites for their stability are achieved. The basic reproduction number and thus the virus transmission coefficient ℜ0 were determined and quantitatively used to study the global stability of the model's steady state. Furthermore, this study considered the sensitivity analysis of the parameters according to ℜ0. The most sensitive variables that are important for infection control are determined using the normalized forward sensitivity index. Data from the United Kingdom collected between May and August 2022, which also aid in demonstrating the usefulness and practical application of the model to the spread of the disease in the United Kingdom, were used. In addition, using the Caputo–Fabrizio operator, Krasnoselskii's fixed point theorem has been used to analyze the existence and uniqueness of the solutions to the suggested model. The numerical simulations are presented to assess the system dynamic behavior. More vulnerability was observed when monkeypox virus cases first appeared recently as a result of numerical calculations. We advise the policymakers to consider these elements to control monkeypox transmission. Based on these findings, we hypothesized that another control parameter could be the memory index or fractional order.
The unexpected breakout and global spread of monkeypox have drawn the attention of scientists due to the continuing COVID-19 pandemic. The prevalence of the largest and most pervasive monkeypox pandemic outside of Africa as of 22 June 2022, is 3,340 confirmed cases reported across the world. In addition to mother-to-child vertical transmission, the monkeypox virus can spread from person to person by direct contact with infectious skin or mucosal skin lesions, respiratory droplets, or indirect contact with contaminated objects or materials. The possibility of community transmission cannot be ruled out, and it may also be sexually transferred by semen or vaginal fluid. The virus that causes monkeypox is called the monkeypox virus, and it is an enveloped, linear, double-stranded DNA virus that belongs to the Chordopoxvirinae subfamily of the Poxviridae family. With symptoms of the disease lasting 2–4 weeks and a death rate that previously ranged from 0 to 11 deaths, monkeypox is often a self-limiting sickness. Intense headaches, fever, lesions, and lymphadenopathy are some of the symptoms of monkeypox. Antiviral medications and smallpox vaccines have been approved for use in various nations in response to the monkeypox outbreak, despite the fact that there is no specific treatment or vaccine for monkeypox virus infection. Before allowing the virus to successfully establish person-to-person transmission, quick action is required to stop the local development of the disease and, consequently, the global monkeypox outbreak (1–11). In Peter et al. (12), modeling and optimal control were used to study monkeypox and the cost-effective strategies were investigated. This study shows that, among all competing measures, combining preventative measures to reduce rodent-to-human disease transmission is the most practical and cost-effective option.
Numerous research articles have been published where both classical and fractional models were constructed, and there is a plethora of literature on modeling infectious diseases. Because fractional-order derivative has unique properties such as heredity and memory that enable it to fully comprehend the dynamics of real phenomena, an analysis based on fractional-order derivative is more advantageous and practical than an analysis based on classical derivative (13, 14). At two separate closed locations, the phenomenon is indistinguishable by the standard derivatives. A generalized derivative known as the fractional order was proposed to address the problems with ordinary derivatives (15). Many researchers used fractional- order derivatives in many fields, as shown in Kumar et al. (16), Higazy et al. (17), Djida and Atangana (18), Baba (19), Owolabi and Atangana (20), Mohammadi et al. (21), Baleanu et al. (22), and Wutiphol and Turab (23). In the realm of mathematical biology, the Mittag–Leffler-type kernel has been used continuously over other derivatives, and numerous epidemiological models, such as for dengue fever, smoking, tuberculosis, measles, Ebola, and other diseases, have been studied using this operator as shown in Asamoah et al. (24), Peter et al. (25, 26), Kumar et al. (27), Morales-Delgadoa et al. (28), Atangana and Baleanu (29), and Atangana et al. (30). Most notably, in Zhang et al. (31), the Mittag–Leffler-type kernel modeling for Ebola–malaria co-infection was investigated by the authors with the best possible control. They strongly recommended the Mittag–Leffler-type kernel. In Kumar et al. (32), investigated the COVID-19 model using singular and non-singular fractional operators and compared the results of these operators. In Aslam et al. (33), the authors examined a recent study on the mathematical modeling of HIV/AIDS using the Mittag–Leffler-type kernel and came to the conclusion that the infection rate decreases with decreasing operator. In Evirgen (34), the authors studied the transmission dynamics of the Nipah virus using the Caputo derivative. One of the interesting segments of their study was to focus on tracing the influence of fractional-order derivatives on the manner in which the model responds. In Ucar (35), the authors investigated a fractional SAIDR model within the framework of the Mittag–Leffler-type kernel. The effectiveness of the fractional operator is shown through a numerical simulation.
Considering the characteristics of exponential decay, the Caputo–Fabrizio fractional-order operator has been preferred over Atangana–Beleanu beta derivatives and a few other operators in the field of mathematical biology with more information (17–19, 24, 36–39). For instance, in Addai et al. (40), the authors studied a novel model of COVID-19 incorporating Alzheimer's disease using the Caputo–Fabrizio fractional-order operator. The results of the aforementioned study revealed that the two diseases have a link and the authors also concluded that the fractional operator is related to the rate of infection. In Shaikh and Nisar (41), the authors also considered the transmission dynamics of a fractional-order typhoid fever model using the Caputo–Fabrizio operator and the existence theory and achieved numerical solutions. In Shah et al. (42), Shah and his co-authors conducted a semi-analytical study of the Pine Wilt Disease (PWD) model with a convex rate via fractional order involving a non-singular kernel. To comprehend the trade-off between the lockdown and the transmission of the virus, Ahmed and his co-authors devised a five-term dynamical system (43). Another use of the Caputo–Fabrizio fractional-order operator was indicated, for instance, in Addai et al. (40), Shaikh and Nisar (41), Shah et al. (42), Ahmed et al. (43), Ullah et al. (44), Abboubakar et al. (45).
Furthermore, in Peter et al. (46), the authors used real data from Nigeria to study the dynamics of the transmission of the monkeypox virus using fractional calculus. The authors presented an argument on the modeling system by studying the infection control policies that will help the public to better understand the significance of control parameters in the eradication of the virus in the studied population. Furthermore, the transmission dynamics of the monkeypox virus was studied using a mathematical modeling approach in Peter et al. (47). In their findings, the authors indicated that the isolation of infected individuals in the human population helps reduce the transmission of the disease, which can serve as a form of intervention to control the spread of the virus.
We observed that none of the studies on the monkeypox virus and its modes of transmission took into account the interaction between the isolated and exposed compartments in the human subpopulation and the results of that contact rate with the rodent population and applied the modeling approach to real data from the United Kingdom. The major goals of this research are to calculate the exponential growth rate of the monkeypox virus, to forecast what might occur in future and how to stop it from spreading, and to understand the effects of non-pharmaceutical intervention on infected individuals, which will be able to guide us on how to deploy intervention resources to contain the spread of the disease. The remaining sections of the article are structured as follows: Section 2 presents some basic definitions and preliminary information, Section 3 presents the model formulation, Sections 4 deals with the dynamism of the model, Section 5 computes the basic reproduction number and some basic mathematical analysis, Section 6 present the endemic equilibrium of the model, Section 7 proves the existence and uniqueness of our model, Section 8 deals with the fitting of the model to real data from the United Kingdom, Section 9 presents numerical schemes and numerical simulations, Section 10 deals with sensitivity analysis, and Section 11 provides some perspectives, discussion, and conclusion.
2. Preliminaries
In this section, we review several key definitions, lemmas, and concepts that are necessary to understand the suggested model.
Definition 2.1 Let f ∈ Q1(p, q), q > p, and α∗ ∈ (0, 1) (17), (40). Then, the Caputo–Fabrizio fractional-order derivative can be defined as
pCFDtαf(t)=G(α)1-α∫ptf′(x)exp[-αt-s1-α]ds.
Here, G(α) is a normalization function, where G(0) = G(1) = 1. The fractional integral of the Caputo–Fabrizio fractional order is defined by:
Lemma 2.2 Assuming there is a function u(t) ∈ Wl[0, η], then the solution of fractional differential equation
{CFDtαf(t)=u(t),t∈[0,η],f(0)=f0,
is given by
f(t)=f0+2(1-α)2(1-α)G(t)f(t)+2α(2-α)G(α)∫0tf(s)ds,t≥0
(24),(17), (40).
Lemma 2.3 Suppose A ⊂ B be a closed convex non-empty subset of A and there exist two operators, T1 and T2, then it is Krasnoselskii's fixed point theorem (40) and it follows that:
T1u + T1u ∈ A, ∀u ∈ A;
T1 is contraction and T2 continuous and compact. Then quantify at least one solution u ∈ A such that
T1u+T2u=u.
3. Model formulation
Using a system of differential equations, we studied both human and rodent populations in a closed homogeneous environment. There are five compartments in a human population of size Nh(t): Susceptible Sh(t); Exposed Eh(t); Infected Ih(t); Isolation/Quarantine Qh(t); and Recovered Rh(t); where Nh(t) = Sh(t) + Eh(t) + Ih(t) + Qh(t) + Rh(t). The rodent population Nr(t) is split into Sr(t) Susceptible; Er(t) Exposed; and Ir(t) Infected. Let Nr(t) = Sr(t) + Er(t) + Ir(t). From the aforementioned description, using the ideas in Yinka-Ogunleye et al. (5), we extend the studies of Peter et al. (46) and (47), then the ordinary differential equations in system (1) describe the dynamics of monkeypox transmission incorporating non-pharmaceutical intervention;
{dShdt=Λh+ξhRh+θhQh-λhSh-μhSh,dEhdt=λhSh-γhEh-ϕhEh-μhEh,dIhdt=ϕhEh-(ψh+μh+νh)Ih,dQhdt=γhEh-(θh+δh+μh+νh)Qh,dRhdt=ψhIh+δhQh-ξhRh-μhRh,dSrdt=Λr-λrSr-μrSrdErdt=λrSr-ϕrEr-μrEr,dIrdt=ϕrEr-(μr+νr)Ir,
where λh=βrhIr+βhhIhNh,λr=βrrIrNr. To capture the memory in the predictions of the monkeypox virus transmission model and also to verify that both sides of the fractional equations have exact dimensions, the time-dependent kernel is defined by the power law correlation function, as in Tilahuna et al. (48); therefore, we propose the following fractional-order model for the monkeypox virus transmission model using the Caputo–Fabrizio fractional-order derivative;
{CFDtαSh(t)=Λh+ξhRh+θhQh-λhSh-μhSh,CFDtαEh(t)=λhSh-γhEh-ϕhEh-μhEh,CFDtαIh(t)=ϕhEh-(ψh+μh+νh)Ih,CFDtαQh(t)=γhEh-(θh+δh+μh+νh)Qh,CFDtαRh(t)=ψhIh+δhQh-ξhRh-μhRh,CFDtαSr(t)=Λr-λrSr-μrSrCFDtαEr(t)=λrSr-ϕrEr-μrEr,CFDtαIr(t)=ϕrEr-(μr+νr)Ir.
The flow diagram of the model equation is presented in Figure 1 while the parameters used in the model and their signification is presented in Table 1.
Transfer diagram of the dynamic transmission of the monkeypox virus.
Interpretation of parameters in the model.
Parameter
Interpretation
Λh
Human recruitment rate
Λr
Rodent recruitment rate
ξh
Immunity loss rate for human
θh
Undetected rate of human after diagnosis
μh, μr
Natural death rate for humans and rodents
νh, νr
Disease-induced death rate for humans and rodents
ϕh, ϕr
The rate at which humans and rodents move from exposed to infectious stage
ψh
The rate of humans recovery from monkeypox
γh
The rate of identifying as suspected case of monkeypox
δh
The rate of moving from isolated to recovered class
βrh
The rate of transmission within rodents and humans
βhh
The rate of transmission within humans
βrr
The rate of transmission within rodents
4. Dynamics of the model
In this section, we focus on the dynamics of the solutions for the suggested models (1) and (2) that are positive, bounded, and invariant. In an epidemiological model, it is important to evaluate the population survival and the expansion that is naturally constrained by scarce resources. As a result, we demonstrate the following theorem.
Theorem 1. The solution of (1) along with initial conditions is positively invariant and bounded in R+8. Therefore,
{limt→∞supSh(t)≤Sh∞=Λh+θhQh∞+ξhRh∞λh+μh,limt→∞supEh(t)≤Eh∞=λhSh∞(ξh+ϕh+μh),limt→∞supIh(t)≤Ih∞=ϕhEh∞(ψh+ηh+μh),limt→∞supQh(t)≤Qh∞=γhEh∞(θh+δh+μh+νh),limt→∞supRh(t)≤Rh∞=ψhIh∞+δhQh∞(ξh+μh),limt→∞supSr(t)≤Sr∞=Λrλr+μr,limt→∞supEr(t)≤Er∞=λrSr∞(ϕr+μr),limt→∞supIr(t)≤Ir∞=ϕrEr∞(νr+μr).
Proof. Using the results in Lin (49) and taking into account the initial values given, from model (2), we obtain
{CFDtαSh(t)|Sh(0)=Λh+ξhRh+θhQh≥0,CFDtαEh(t)|Eh(0)=λhSh≥0,CFDtαIh(t)|Ih(0)=ϕhEh≥0,CFDtαIh(t)|Qh(0)=γhEh≥0,CFDtαRh(t)|Rh(0)=ψhIh+δhQh≥0,CFDtαSv(t)|Sr(0)=Λr≥0,CFDtαEv(t)|Er(0)=λrSr≥0,CFDtαIv(t)|Ir(0)=ϕrEr≥0.
From Equation (4), we can see that Sh(0) > 0, Eh(0) > 0, Ih(0) > 0, Rh(0) > 0, Sv(0) > 0, Ev(0) > 0, Iv(0) > 0, for all t > 0. From Equation (2), the first equation gives
CFDtαSh(t)≤Λh+ξhRh+θhQh-λhSh-μhSh≥0.
Then, by applying the fractional comparison technique, we obtain the first estimate of Equation (4). We continue for the second equation of the system of Equation (2), we obtain
CFDtαEh(t)≤λhSh-γhEh-ϕhEh-μhEh≥0.
Therefore, we get the second estimate of Equation (1). We continue again for the third equation of the system of Equation (2), we obtain
CFDtαIh(t)≤ϕhEh-(ψh+μh+νh)Ih≥0,
and, consequently, we obtain the third estimate of Equation (4). Similarly, for the fourth to eighth equation, we obtain the estimate of Equation (4). Hence, Theorem 1 is complete.
4.1. Monkeypox equilibrium state
The monkeypox model is studied by obtaining the equilibrium states. To verify the existence of the equilibrium points, the derivatives of the model on the right-hand side are set to zero, which provides the monkeypox disease free equilibrium points.
We assume Eh, Er, Ih, Ir, Qh, Rh, Sh, Sr be the solution to the monkeypox model with the initial condition in a feasible region such that
Γh=Eh,Ih,Qh,Rh,Sh∈ℝ5:Nh=Λhμh,Γr=Er,Ir,Sr∈ℝ3:Nr=Λrμr,
where the human population is represented as
Nh=Eh(t)+Ih(t)+Qh(t)+Rh(t)+Sh(t),
and the rodent population, respectively,
Nr=Er(t)+Ir(t)+Sr(t).
To achieve the disease-free equilibrium state, the derivatives are set to zero as seen in (10) to obtain
E*=(Eh*,Er*,Ih*,Ir*,Qh*,Rh*,Sh*,Sr*).
By setting the derivatives to zero, we obtain
dEhdt=dErdt=dIhdt=dIrdt=dQhdt=dRhdt=dShdt=dSrdt=0;
hence, Equation (9) is represented as
E*=(0,0,0,0,0,0,Λhμh,Λrμr).
This equation describes a population free of monkeypox infection and is denoted as E*
5. The basic reproduction number
We derive the basic reproduction number ℜ0 by using the next-generation matrix approach (25). Since Eh, Ih, Qh, and Ir are the disease-infected classes, hence,
f=(0λhSh000000),v=(-Λh-ξhRh-θhQh+λhSh+μhShγhEh+ϕhEh+μhEh-ϕhEh+(ψh+μh+νh)Ih-γhEh+(θh+δh+μh+νh)Qh-ψhIh-δhQh+ξhRh+μhRh-Λr+λrSr+μrSr-λrSr+ϕrEr+μrEr-ϕrEr+(μr+νr)Ir).F=(00βhhΛhμh0βhhΛhμh000000000000000),V=(γh+ϕh+μh000-ϕhψh+μh+νh00-γh0θh+δh+μh+νh0000μr+νr).V-1=[1γ1+μ1+ϕ1000ϕ1γ1μ1+γ1ν1+γ1ψ1+μ2+ν1μ1+μ1ϕ1+μ1ψ1+ν1ϕ1+ψ1ϕ11ν1+μ1+ψ100γ1γ1δ1+δ1μ1+δ1ϕ1+γ1μ1+ν1γ1+γ1θ1+μ12+μ1ν1+μ1ϕ1+μ1θ1+ν1ϕ1+θ1ϕ101ν1+μ1+θ1+δ100001ν2+μ2].
The next-generation matrix (G) is given by
G=F.V-1=[β1λ1ϕ1Nhμ1(γ1μ1+γ1ν1+γ1ψ1+μ2+ν1μ1+μ1ϕ1+μ1ψ1+ν1ϕ1+ψ1ϕ1)β1λ1Nhμ1(ν1+μ1+ψ1)0β1λ1Nhμ1(ν1+μ1)000000000000].
The basic reproduction number ℜ0 is the dominant eigenvalue (spectral radius) of the next-generation matrix G, that is, ℜ0 = ρ(G)
ℜ0=βhhΛhϕhμh(γh+ϕh+μh)(ψ1+μh+νh)
5.1. Stability of monkeypox-free equilibrium (MFE)
Investigating the stability of the monkeypox disease-free equilibrium, we compute the Jacobian matrix of the system at the disease-free equilibrium by obtaining the eigenvalues, which will be used to determine the stability of the model.
JE*=(-βhhIh+βrhIrNh-μh00θhξh00-(βhh+βrh)ShNhβhhIh+βrhIrNhζ100000(βhh+βrh)ShNh0φh0000000γh0ζ20000000δh-μh-ξh00000000-βrrIrNr-μr0-βrrSrNr00000βrrIrNr-φr-μrβrrSrNr000000φr-μr-νr),
where ζ1 and ζ2 are represented in Equations (17) and (18)
ζ1=-γh-φh-μh,ζ2=-θh-δh-μh-νh.
Evaluating JE* at the monkeypox-free equilibrium (MFE), we obtain
JMFE*=(-μh00θhξh00-(βhh+βrh)ΛhNhμh0-γh-φh-μh00000(βhh+βrh)ΛhNhμh0φh0000000γh0-θh-δh-μh-νh0000000δh-μh-ξh00000000-μr0-βrrΛrNrμr000000-φr-μrβrrΛrNrμr000000φr-μr-νr).
We compute the eigenvalues from the JMFE* using the characteristic polynomial of O8, which will not be represented as a result of its lengthiness. The eigenvalues and characteristic polynomial are calculated by |JMFE*-I|, where I is an 8 × 8 unit matrix, and the values of λ are obtained:
λ=(0-μh-μh-ξh-μr-γh-φh-μh-θh-δh-μh-νh-2Nrμr2+(-νr-φr)Nrμr+μr(Nr(νr-φr)2μr+4Λrβrrφr)Nr2Nrμr-2Nrμr2+(-νr-φr)Nrμr-μr(Nr(νr-φr)2μr+4Λrβrrφr)Nr2Nrμr)
Let Δ1and Δ2 be well represented from Equation (20) in Equations (21) and (22)
Δ1=μr(Nr(νr-φr)2μr+4Λrβrrφr)Nr,Δ2=2Nrμr2+(-νr-φr)Nrμr.
Then,
λ1=0,λ2=-μh,λ3=-(μh+ξh),λ4=-μrλ5=-(γh+φh+μh),λ6=-(θh+δh+μh+νh),λ7=-Δ2+Δ12Nrμr,λ8=-Δ2-Δ12Nrμr.
From the calculated eigenvalues, we obtain negative real parts, that is, the monkeypox-free equilibrium is asymptotically stable if
-Δ2-Δ12Nrμr<0.
Upon simplification, we obtain Equation (31):
Nrμr(2μr-νr-φr)2Nrμr(νr-φr)2+4Λrβrrφr<1.
Therefore, the monkeypox-free equilibrium state is asymptotically stable.
5.2. Global stability of the equilibrium state
If ℜ0 < 1, then the monkeypox-free equilibrium is globally asymptotically stable; otherwise, it is unstable. This is proven by the Lyapunov function such that
L(Eh)=Eh
Differentiating, we obtain
L′(Eh)=Eh′=λhSh-γhEh-ϕhEh-μhEh=λhSh-(γh+ϕh+μh)Eh.
At the disease-free equilibrium state as seen in Equation (11), Sh=Λhμh,L′(Eh)=λh(Λhμh)-(γh-ϕh-μh)EhEh′=(γh+ϕh+μh)[Λhλhμh(γh+ϕh+μh)Eh-1]EhEh′=(γh+ϕh+μh)(ℜ0-1)Eh≤0 if ℜ0≤0.
From the result obtained in Equation (39), we can see that Eh′≤0 provided ℜ0 ≤ 0 as well as Eh′=0 provided that ℜ0 = 0 or Eh = 0. Global stability of the disease-free equilibrium is asymptotically stable, if ℜ0 ≤ 0; otherwise, it is unstable.
6. Endemic equilibrium state
The endemic equilibrium state occurs when the rate of infection persists in the population and it is represented in Equations (40 - 47) by Eh**,Er**,Ih**,Ir**,Qh**,Rh**,Sh**,Sr**.Eh**=[μh3+k1μh2+(k2+k3)μh+k4]Λhλhμh5+p1μh4+p2·μh3+p3·μh2+μh·p4+λhνhξh·p5Er**=λrΛr(μr+φr)(μr+λr)Ih**=(μh2+(δh+νh+θh+ξh)μh+δhξh+νhξh+θhξh)Λhλhφhμh5+p1μh4+p2·μh3+p3·μh2+μh·p4+λhνhξh·p5+p6Ir**=φrλrΛrλrμr2+λrμrνr+λrμrφr+λrνrφr+μr3+μr2νr+μr2φr+μrνrφrQh**=(γhμh2+γh(νh+ψh+ξh)μh+γh(νhξh+ψhξh))Λhλhμh5+p1μh4+p2·μh3+p3·μh2+μh·p4+λhνhξh·p5+p6Rh**=(δhψh+μhψh+νhψh+ψhθh)λhΛhφh+(δhγhμh+δhγhνh+δhγhψh)λhΛhμh5+p1μh4+p2·μh3+p3·μh2+μh·p4+λhνhξh·p5+p6Sh**=Λhμh4+Λh·h1·μh3+Λh·h2·μh2+Λh·h3·μh+Λh·h4μh5+p1μh4+p2·μh3+p3·μh2+μh·p4+λhνhξh·p5+p6Sr**=Λrλr+μr,
where
d1=(ψh+νh),d2=(δh+νh+θh+ξh),d3=(δh+νh+θh),k1=(ψh+2νh+δh+θh+ξh)k2=d1+d2,k3=ξh·d3,k4=d1·ξh·d3,p1=δh+γh+λh+2νh+ψh+θh+φh+ξh,p2=δhγh+δhλh+δhνh+δhψh+δhφh+δhξh+γhλh+2γhνh+γhψh+γhθh+γhξh+2λhνh+λhψh+λhθh+λhφh+λhξh+νh2+νhψh+νhθh+2νhφh+2νhξh+ψhθh+ψhφh+ψhξh+θhφh+θhξh+φhξh,p3=δhγhλh+δhγhνh+δhγhψh+δhγhξh+δhλhνh+δhλhψh+δhλhφh+δhλhξh+δhνhφh+δhνhξh+δhψhφh+δhψhξh+δhφhξh+2γhλhνh+γhλhψh+γhλhξh+γhνh2+γhνhψh+γhνhθh+2γhνhξh+γhψhθh+γhψhξh+γhθhξh+λhνh2+λhνhψh+λhνhθh+2λhνhφh+2λhνhξh+λhψhθh+λhψhφh+λhψhξh+λhθhφh+λhθhξh+λhφhξh+νh2φh+νh2ξh+νhψhφh+νhψhξh+νhθhφh+νhθhξh+2νhφhξh+ψhθhφh+ψhθhξh+ψhφhξh+θhφhξh,p4=δhγhλhνh+δhγhλhψh+δhγhνhξh+δhγhψhξh+δhλhνhφh+δhλhνhξh+δhλhψhφh+δhλhψhξh+δhλhφhξh+δhνhφhξh+δhψhφhξh+γhλhνh2+γhλhνhψh+2γhλhνhξh+γhλhψhξh+γhνh2ξh+γhνhψhξh+γhνhθhξh+γhψhθhξh+λhνh2φh+λhνh2ξh+λhνhψhφh+λhνhψhξh+λhνhθhφh+λhνhθhξh+2λhνhφhξh+λhψhθhφh+λhψhθhξh+λhθhφhξh+νh2φhξh+νhψhφhξh+νhθhφhξh+ψhθhφhξh,p5=δhφh+γhνh+γhψh+νhφh+θhφh,p6=δhλhνhφhξh+γhλhνh2ξh+γhλhνhψhξh+λhνh2φhξh+λhνhθhφhξh,h1=δh+γh+2νh+ψh+θh+φh+ξh,h2=δhγh+δhνh+δhψh+δhφh+δhξh+2γhνh+γhψh+γhθh+γhξh+νh2+νhψh+νhθh+2νhφh+2νhξh+ψhθh+ψhφh+ψhξh+θhφh+θhξh+φhξh,h3=δhγhνh+δhγhψh+δhγhξh+δhνhφh+δhνhξh+δhψhφh+δhψhξh+δhφhξh+γhνh2+γhνhψh+γhνhθh+2γhνhξh+γhψhθh+γhψhξh+γhθhξh+νh2φh+νh2ξh+νhψhφh+νhψhξh+νhθhφh+νhθhξh+2νhφhξh+ψhθhφh+ψhθhξh+ψhφhξh+θhφhξh,h4=δhγhνhξh+δhγhψhξh+δhνhφhξh+δhψhφhξh+γhνh2ξh+γhνhψhξh+γhνhθhξh+γhψhθhξh+νh2φhξh+νhψhφhξh+νhθhφhξh+ψhθhφhξh,
7. Existence and uniqueness results for the monkeypox transmission model with non-pharmaceutical intervention
We reformulate Equation (2) as follows:
{Φ1(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=Λh+ξhRh+θhQh-λhSh-μhSh,Φ2(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=λhSh-γhEh-ϕhEh-μhEh,Φ3(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=ϕhEh-(ψh+μh+νh)Ih,Φ4(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=γhEh-(θh+δh+μh+νh)QhΦ5(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=ψhIh+δhQh-ξhRh-μhRh,Φ6(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=Λr-λrSr-μrSrΦ7(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=λrSr-ϕrEr-μrEr,Φ8(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t))=ϕrEr-(μr+νr)Ir.
From Equation (10), the developed model of Equation (1) can be written in the form
{CFDtαΦ(t)=ϒ(t,Φ(t)),t∈[0,η],0<α≤1,Φ(0)=Φ0,Φ(t)={Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t),Φ0={Sh(0),Eh(0),Ih(0),Qh(0),Rh(0),Sr(0),Er(0),Ir(0),
therefore,
ϒ(t,Φ(t))={Φ1(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)),Φ2(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)),Φ3(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)),Φ4(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)),Φ5(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)),Φ6(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)),Φ7(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)),Φ8(t,Sh(t),Eh(t),Ih(t),Qh(t),Rh(t),Sr(t),Er(t),Ir(t)).
With the help of Lemma 2.4, Equation (49) yields
{Φ(t)=Φ0(t)+2(1-α)2(1-α)G(α)ϒ(t,Φ(t))+2α(2-α)G(α)×∫0tϒ(s,Φ(s))ds.
Furthermore, let ussay E = C([0, η]) is the Banach space, and supposing that the following assumptions hold;
(H1), there exists a non-negative constant Q, W, and k ∈ [0, 1) such that
ϒ(t,Φ(t))≤Q|Φ|k+W.
(H2) There exists a nonnegative constant Cρ > 0 for all Φ,Φ~∈E, then
|ϒ(t,Φ(t))-ϒ(t,Φ~(t))|≤Cρ[|Φ-Φ~|].
Furthermore, let us define operator Am : E → E such that
Amℵ(t)=M1Φ(t)+M2Φ(t),
therefore, we can see that
{M1Φ(t)=Φ0(t)+2(1-α)2(1-α)G(α)ϒ(t,Φ(t)),M2Φ(t)=2α(2-α)G(α)∫0tϒ(s,Φ(s))ds.
From this knowledge, Equation (52) can be written as
{AmΦ(t)=Φ0(t)+2(1-α)2(1-α)G(α)ϒ(t,Φ(t))+2α(2-α)G(α)×∫0tϒ(s,Φ(s))ds.
Theorem 2. Suppose that (H1) and (H2) hold, such that 2(1-α)2(1-α)G(α)Cρ<1, then, the monkeypox transmission model with non-pharmaceutical intervention has at least one solution.
Proof. For simplicity, we divide the proof into two steps.
Step 1. We prove that operator M1 is contraction. Then, let Φ~∈Ω, where Ω = {Φ ∈ Z : ||Φ|| ≤ ϑ, ϑ > 0} is a close convex set, thus
|M1Φ(t)-M2Φ|=2(1-α)2(1-α)G(α)maxα∈[0,η]|ϒ(t,Φ(t))-ϒ(t,Φ~(t))|,≤2(1-α)2(1-α)G(α)Cρ||Φ-Φ~||.
Thus,
||M1Φ-M2Φ(t)||≤2(1-α)2(1-α)G(α)Cρ||Φ-Φ~||.
Hence, M1 is contraction since 2(1-α)2(1-α)G(α)Cρ<1.
Step 2. We also prove that M2 is compact and also continuous; for all Φ ∈ Ω, then M2 will be continuous as Φ is continuous, thus
||M2(Φ)||=maxt∈[0,η]|2α(2-α)G(α)∫0tϒ(s,Φ(s))ds|,≤2α(2-α)G(α)η∫0t|ϒ(s,Φ(s))|ds.≤2α(2-α)G(α)η[Q|Φ|k+W].
Hence, M2 is boundedness. For equicontinuous, let t1, t2 ∈ [0, η] such that
|(M2Φ)(t1)-(M2Φ)(t2)|=2α(2-α)G(α)maxt∈[0,η]|∫0t1ϒ(s,Φ(s))ds-∫0t2Φ(s,ℵ(s))ds|≤2α(2-α)G(α)[Q|Φ|k+W]|t1-t2|.
As t1 → t2, then |(M2Φ)(t1) − (M2Φ)(t2)| → 0, which makes operator M2 equicontinuous and compact by the Arzela–Ascoli theorem. Therefore, by Lemma 2.3, the existence for the monkeypox transmission model with non-pharmaceutical intervention has at least one solution. □
Theorem 3. Suppose that ∃ is a nonnegative integer Λρ is > 0 such that
Λρ=[2(1-α)2(1-α)G(α)Lρ+2α(2-α)G(α)ηLρ]<1,
then operator Am has a unique fixed point.
Proof. Let Φ,Φ~∈Ω, then we say
||AmΦ-AmΦ~||≤||M1Φ-M1Φ~||+||M2Φ-M2Φ~||,≤2(1-α)2(1-α)G(α)maxt∈[0,η]|ϒ(t,Φ(t))-ϒ(t,Φ~(t))|+2α(2-α)G(α)maxt∈[0,η]|∫0tϒ(s,Φ(s))ds-∫0tϒ(s,Φ~(s))ds|≤[2(1-α)2(1-α)G(α)Cρ+2α(2-α)G(α)ηCρ]||Φ-Φ~||,=Λρ||Φ-Φ~||.
Hence, by the Banach contraction principle, Am has a unique fixed point. Consequently, the monkeypox transmission model with non-pharmaceutical intervention has a unique solution. □
8. Fitting of model to data
We used the available public database to collect our data while the formulated model of Equation (1) includes 16 parameters. To treat the waggliness of the reported daily new cases, we smoothed the data to remove noise from the data set so as to make it suitable for our analysis. The total population of the United Kingdom is 68,530,739 (1), which was used for calculating the initial number of susceptible humans, while the initial value for the number of infected humans was calculated from the reported daily new cases. Other initial values were assumed.
The link to the data used for this research and the initial values; Sh(t) = 68530739; Eh(t) = 0; Ih(t) = 31412; Qh(t) = 0; Sr(t) = 1074103; Er(t) = 1074103; and Ir(t) = 1074103; can be found in the Data Availability section. The parameters are fitted based on the smoothed reported daily new cases of infected humans from May to August 2022. This information was taken from the United Kingdom public health database (1). The nonlinear least square technique was used to fit the model using python programming. Table 2 shows all of the parameter values that were fitted, and Figure 2 shows the data fitting of the observed smoothed daily new cases.
Parameter values in the model.
Parameter
Value
Source
Λh
8644
Estimated
Λr
0.9
Assumed
ξh
0.00001
Fitted
θh
0.029
Fitted
μr
0.00200
(46)
μh
0.05
(1)
νh, νr
0.00008, 0.0001
Fitted
ϕh, ϕr
0.007
Fitted
ψh
0.056
Fitted
γh
0.0081
Fitted
δh
0.012
Fitted
βrh
0.000009
Fitted
βhh
0.00008
Fitted
βrr
0.0057
Fitted
Model fitting.
9. Numerical scheme
In this section, we present the numerical results for the monkeypox transmission model with non-pharmaceutical intervention based on the Lagrange interpolation. Details about the numerical scheme is presented in Atangana and Owolabi (50). The Cauchy problem of the CF fractional derivative can be given as:
CFDtαΦ(t)=ϒ(t,Φ(t)),
On the other hand, we can express Equation (60) as
Φ(t)=Φ0(t)+(1-α)G(α)ϒ(t,Φ(t))+αG(α)×∫0tϒ(s,Φ(s))ds.
Taking Equation (61) at the point tn+1 = (n + 1)h and tn = nh, n = 0, 1, 2, 3, …, with h being the time step, we have
Φ(tn+1)=Φ(0)+(1-α)G(α)ϒ(tn,Φ(tn))+αG(α)×∫tntn+1ϒ(s,Φ(s))ds,Φ(tn)=Φ(0)+(1-α)G(α)Φ(tn-1,ℵ(tn-1))+αG(α)×∫tntn+1ϒ(s,Φ(s))ds.
Taking the results of Equations (62)-(63) in
Φ(tn+1)-ℵ(tn)=(1-α)G(α)(ϒ(tn,Φ(tn))-ϒ(tn-1,Φ(tn-1)))+αG(α)×∫tntn+1ϒ(s,Φ(s))ds,
Equation (64) in the two-step Lagrange polynomial gives
Φ(tn+1)-Φ(tn)=(1-α)G(α)(ϒ(tn,Φ(tn))-ϒ(tn-1,Φ(tn-1)))+αG(α)×∫tαtα+1[ϒ(tn,Φ(tn))h(s-tn-1)-ϒ(tn-1,Φ(tn-1))h(s-tn)]ds.
The aforementioned Equation (65) leads to
Φ(tn+1)-Φ(tn)=(1-α)G(α)(ϒ(tn,Φ(tn))-ϒ(tn-1,Φ(tn-1)))+αG(α)×[ϒ(tn,Φ(tn))h∫tntn+1(s-tn-1)ds-ϒ(tn-1,Φ(tn-1))h∫tntn+1(s-tn)ds].
Solving the integrals in Equation (66) yields
∫tntn+1(s-tn-1)ds=32h2,∫tntn+1(s-tn)ds=12h2.
Substituting Equation (67) into Equation (66), then generalizing the numerical scheme of CF is as follows:
Φn+1=ℵn+[(1-α)G(α)+3hα2G(α)]Φ(tn,Φn)-[(1-α)G(α)+hα2G(α)]Φ(tn-1,ℵn-1).
Thus, in terms of our CF-fractional monkeypox transmission model with non-pharmaceutical intervention, we obtain;
Shn+1=Shn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Shn)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Shn-1).Ehn+1=Ehn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Ehn)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Ehn-1).Ihn+1=Ihn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Ihn)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Ihn-1).Qhn+1=Qhn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Qhn)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Qhn-1).Rhn+1=Shn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Rhn)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Rhn-1).Srn+1=Shn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Srn)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Srn-1).Ern+1=Shn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Ern)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Ern-1).Irn+1=Shn+[(1-α)G(α)+3hα2G(α)]ϒ(tn,Irn)-[(1-α)G(α)+hα2G(α)]ϒ(tn-1,Irn-1).
10. Sensitivity analysis
Since an epidemiological system's parameters are either estimated or fitted, there is some degree of uncertainty in the numbers that are utilized to derive conclusions about the underlying epidemic. It is crucial to evaluate the individual effects of each parameter on the dynamics of the epidemic to identify those effects that have the greatest impact on the epidemic's spread or contraction. For biological factors included in the proposed monkeypox model, we perform the sensitivity analysis in this section. This analysis is investigated analytically by computing ∂ℜ0∂p, where, p = (βhh, Λh, ϕh, μh, γh, νh, and ψh). The sensitivity of ℜ0 to each parameter is as follows:
∂ℜ0∂βhh=λhϕhμh(γh+μh+ϕh)(νh+μh+ψh)>0,∂ℜ0∂Λh=βhhϕhμh(γh+μh+ϕh)(νh+μh+ψh)>0,∂ℜ0∂ϕh=βhhΛh(γh+μh)μh(γh+μh+ϕh)(νh+μh+ψh),∂ℜ0∂μh=-βhhΛh(μh(γh+μh+ϕh)+μh(νh+μh+ψh)+(γh+μh+ϕh)(νh+μh+ψh))μh2(γh+μh+ϕh)2(νh+μh+ψh)2<0,∂ℜ0∂γh=-βhhλhϕhμh(γh+μh+ϕh)2(νh+μh+ψh)<0,∂ℜ0∂νh=-βhhλhϕhμh(γh+μh+ϕh)2(νh+μh+ψh)2<0,∂ℜ0∂ψh=-βhhλhϕhμh(γh+μh+ϕh)2(νh+μh+ψh)2<0,
thus,
∂ℜ0∂βhh=124645.995943846∂ℜ0∂Λh=16.9011519923859∂ℜ0∂ϕh=0.1746837421029776(0.008139+ψh)2∂ℜ0∂μh=-9.912×10-7μh2+4.7035744×10-8μh2+2.797853632×10-10μh2(μh4+0.014236μh3+0.00012055158688μ1+7.17083788864×10-7),∂ℜ0∂γh=-0.15096125860751(γh+0.007039)2,∂ℜ0∂νh=-0.59600691709814(νh+0.056039)2,∂ℜ0∂ψh=-0.59600691709814(νh+0.056039)2.
The sensitivity index technique will help measure the most sensitive parameters for the fundamental reproductive number ℜ0 (Borgonovo et al. (51) for details about the method). The fundamental reproduction number's normalized sensitivity index is provided by Spℜ0=∂ℜ0∂p.pℜ0, where p is a parameter as defined earlier. We obtain
Sβhhℜ0=1,SΛhℜ0=1,Sϕhℜ0=γh+μhγh+μh+ϕh,Sμhℜ0=μh(γh+μh+ϕh)+μh(νh+μh+ψh)+(γh+μh+ϕh)(νh+μh+ψh)(γh+μh+ϕh)(νh+μh+ψh),Sγhℜ0=γhγh+μh+ϕh,Sνhℜ0=νhνh+μh+ψh,Sψhℜ0=ψhνh+μh+ψh.
The sensitivity indices using the parameter values given in Table 2 are presented in Table 3. The sensitivity analysis of βhh, Λh, ϕh, ψh, νh, γh, and μh with respect to ℜ0 and their graphs are presented in Figure 3.
The sensitivity index of ℜ0 with respect to parameter p of the system (1).
Parameter
Sensitivity index
νh
–0.0014
Λh
1
ϕh
–1.003
μh
–1.003
βhh
1
γh
–0.5350
ψh
–0.9979
The sensitivity analysis of ℜ0 with respect to the parameter p of the system (1).
Two of the sensitivity indices are positive while others are negative, as can be seen in Table 3. Additionally, the majority of these indices are functions of the Caputo–Fabrizio fractional monkeypox model parameters. This implies that changing one of the parameters slightly will alter the dynamics of the epidemic. The basic reproductive number ℜ0 normalized sensitivity indices to the Caputo–Fabrizio fractional monkeypox model parameters are calculated. We conclude that increasing the rate of recovery and the rate of identifying suspected cases, that is, isolation and quarantining of the monkeypox virus carrier will aid in decreasing the ℜ0, which is an affirmation of the effect of non-pharmaceutical intervention to combat the spread of the virus.
11. Discussion and conclusion
Following the estimation of parameter values and data fitting, we simulate the Caputo–Fabrizio fractional monkeypox virus model using the parameter values, as presented in Table 2. The fitted Caputo–Fabrizio curve and ℜ0 are given in Figure 2. Figures 4, 5 show dynamic behavior for all the nine compartments involved in the proposed Caputo–Fabrizio fractional monkeypox virus model. We observed a significantly high susceptibility and infection in the solution pathways of individual species. The work of Hammouch et al. (52), Bonyah et al. (53), Peter (54), and Sene (55) have provided a strong basis for the discussion of our results. This indicates that, whenever the memory index increases, the rate at which people get infected with monkeypox virus reduces and vice versa, which then indicates that, using fractional order, we can obtain clear qualitative information on monkeypox virus transmission. In Figure 6, we varied the input parameter γh on quarantine and exposed, respectively, to observe variation in the system dynamics. We noticed the contribution of this parameter in the transmission pathways of infected individuals. In a similar way, we varied the input parameters δh and ψh on individual recovery and noticed the variation in the trajectory of monkeypox recovery. We discovered that the rate at which humans and rodents move from exposed to infectious stage is also important and potentially dangerous in terms of increasing the level of monkeypox infection.
Numerical trajectory of the CF-fractional-order derivative, α, of Equation (2). (A) Dynamics of susceptible (Sh) humans class. (B) Dynamics of expose (Eh) humans class. (C) Dynamics of infected (Ih) humans class. (D) Dynamics of quarantine (Qh) class.
Numerical trajectory of the CF-fractional-order derivative, α, of Equation (2). (A) Dynamics of recovery (Rh) class. (B) Dynamics of susceptible (Sr) rodents class. (C) Dynamics of exposed (Er) rodents class. (D) Dynamics of asymptomatic infected (Ir) rodents class.
Numerical trajectory of varying δh, γh, ψh, and ϕh when α = 0.95. (A) Variation of δh on recovery class. (B) Variation of γh on expose class. (C) Variation of γh on quarantine class. (D) Variation of ψh on recovery class. (E) Variation of ϕh on infected class.
In conclusion, we provided a brief overview of the monkeypox virus and the dynamics of its transmission in this study. We investigated the spread of monkeypox virus and its effect on non-pharmaceutical intervention, thus quarantine. Positiveness, invariance, boundedness, and equilibrium points of the solutions are thus examined as fundamental features of mathematical models. We considered real data of the monkeypox virus from the United Kingdom, and the best fit curve has been obtained (see Figure 2). As a result, we created a novel, dimensionally consistent Caputo–Fabrizio fractional-order model. Krasnoselskii's fixed point theorem has been used to demonstrate that the system has a solution. The Adams–Bashforth method has been used to display numerical simulations of the suggested pandemic model for various fractional orders and parameter values. We looked into the impact of factors on the expansion and contraction of the quarantine compartment, recovery compartment, and infected compartment on the spread and regression of the pandemic with the use of numerical simulations. As can be inferred from the data, it is clear that the fractional-order equations can help explain this unique effect of the monkeypox. Real-world data can be used to test the accuracy of a mathematical model that has been created. The key challenge, however, is where to find these data and/or how to obtain the right curve for the collected data. The mathematical representation of the monkeypox has been the subject of numerous studies. To the best of our knowledge, there is still no research on fractional modeling that uses actual data on the monkeypox in the United Kingdom. Using actual data on the monkeypox from the United Kingdom, a fractional-order modeling has been shown in this study. The numerical results of this study show that the spread of monkeypox can be stopped if the number of contacts with infected people can be decreased through methods such as effective mass education, improved quarantine facilities, or increased testing of the general population, that is, performing routine tests not only on exposed individuals but also on those who have come into contact with infected patients. As a result, these studies offer other professionals and scientists who focus on infectious diseases insight that may help them in future to control the outbreak of monkeypox and contribute to the development of further treatment options. This study may provide insight into potential future research projects in this regard. Future study of the monkeypox can take into account other fractional operator types, both with and without single kernels. Furthermore, data imputation techniques can be used to fit rodent population parameters from the number of monkeypox disease since the number of rodents cannot be determined.
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
KO, EA, and MN: conceptualization. EA, AA, UA, and KO: methodology, software, and formal analysis. EA, UA, and KO: validation, investigation, and visualization. MN and AA: resources. EA, MN, UA, and KO: data curation. EA and KO: writing and original draft preparation, writing, reviewing, and editing. KO and MN: supervision and project administration. All authors have read and agreed to the published version of the manuscript.
Authors would like to appreciate the Black in Mathematics Association (BMA) and the Human Sciences Research Council (HSRC) for giving us the platform to collaborate as young researchers to carry out this work.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher's note
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